Bot GPT 4-o mini To rewrite \(\sqrt[3]{-16}\) as a product of an integer and another irrational cube root, we can first factor \(-16\) into its components.
We note that: \[ -16 = -1 \cdot 16 \]
Next, we can express \(16\) as \(2^4\): \[ -16 = -1 \cdot 2^4 \]
Now we can rewrite the cube root: \[ \sqrt[3]{-16} = \sqrt[3]{-1 \cdot 2^4} \]
Using the property of cube roots, this can be simplified to: \[ \sqrt[3]{-16} = \sqrt[3]{-1} \cdot \sqrt[3]{2^4} \]
Calculating each part:
- \(\sqrt[3]{-1} = -1\)
- \(\sqrt[3]{2^4} = \sqrt[3]{2^3 \cdot 2} = \sqrt[3]{2^3} \cdot \sqrt[3]{2} = 2 \cdot \sqrt[3]{2}\)
Now, substituting these values back in: \[ \sqrt[3]{-16} = -1 \cdot (2 \cdot \sqrt[3]{2}) = -2 \cdot \sqrt[3]{2} \]
Therefore, we can rewrite \(\sqrt[3]{-16}\) as: \[ \sqrt[3]{-16} = -2 \sqrt[3]{2} \]
Thus, \(\sqrt[3]{-16}\) rewritten as a product of an integer and another irrational cube root is: \[ -2 \sqrt[3]{2} \]
1 answer